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On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system

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  • We study the relations between the global dynamics of the 3D Leray-$\alpha $ model and the 3D Navier-Stokes system. We prove that time shifts of bounded sets of solutions of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and $ \alpha $ approaches zero. In particular, we show that the trajectory attractor of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system when $\alpha \rightarrow 0\+.$
    Mathematics Subject Classification: Primary: 35Q30, 37L30; Secondary: 76D03, 76F20, 76F55, 76F65.

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